Optimal. Leaf size=266 \[ \frac{d \left (-\tan ^2(e+f x)\right )^{\frac{1-n}{2}+\frac{n-1}{2}} (d \tan (e+f x))^{n-1} \left (-\frac{b (1-\sec (e+f x))}{a+b \sec (e+f x)}\right )^{\frac{1-n}{2}} \left (\frac{b (\sec (e+f x)+1)}{a+b \sec (e+f x)}\right )^{\frac{1-n}{2}} F_1\left (1-n;\frac{1-n}{2},\frac{1-n}{2};2-n;\frac{a+b}{a+b \sec (e+f x)},\frac{a-b}{a+b \sec (e+f x)}\right )}{a f (1-n)}-\frac{d \left (-\tan ^2(e+f x)\right )^{\frac{1-n}{2}+\frac{n+1}{2}} (d \tan (e+f x))^{n-1} \text{Hypergeometric2F1}\left (1,\frac{n+1}{2},\frac{n+3}{2},-\tan ^2(e+f x)\right )}{a f (n+1)} \]
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Rubi [F] time = 0.0498594, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{(d \tan (e+f x))^n}{a+b \sec (e+f x)} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{(d \tan (e+f x))^n}{a+b \sec (e+f x)} \, dx &=\int \frac{(d \tan (e+f x))^n}{a+b \sec (e+f x)} \, dx\\ \end{align*}
Mathematica [B] time = 4.5795, size = 786, normalized size = 2.95 \[ \frac{2 \tan \left (\frac{1}{2} (e+f x)\right ) (d \tan (e+f x))^n \left ((a+b) F_1\left (\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-b F_1\left (\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),\frac{(a-b) \tan ^2\left (\frac{1}{2} (e+f x)\right )}{a+b}\right )\right )}{f (a+b \sec (e+f x)) \left (\sec ^2\left (\frac{1}{2} (e+f x)\right ) \left ((a+b) F_1\left (\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-b F_1\left (\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),\frac{(a-b) \tan ^2\left (\frac{1}{2} (e+f x)\right )}{a+b}\right )\right )-\frac{2 (n+1) \tan ^2\left (\frac{1}{2} (e+f x)\right ) \sec ^2\left (\frac{1}{2} (e+f x)\right ) \left ((a+b)^2 \left (F_1\left (\frac{n+3}{2};n,2;\frac{n+5}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-n F_1\left (\frac{n+3}{2};n+1,1;\frac{n+5}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )+b n (a+b) F_1\left (\frac{n+3}{2};n+1,1;\frac{n+5}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),\frac{(a-b) \tan ^2\left (\frac{1}{2} (e+f x)\right )}{a+b}\right )+b (a-b) F_1\left (\frac{n+3}{2};n,2;\frac{n+5}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),\frac{(a-b) \tan ^2\left (\frac{1}{2} (e+f x)\right )}{a+b}\right )\right )}{(n+3) (a+b)}+2 n \tan \left (\frac{1}{2} (e+f x)\right ) \csc (e+f x) \sec (e+f x) \left ((a+b) F_1\left (\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-b F_1\left (\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),\frac{(a-b) \tan ^2\left (\frac{1}{2} (e+f x)\right )}{a+b}\right )\right )-16 n \sin ^5\left (\frac{1}{2} (e+f x)\right ) \cos \left (\frac{1}{2} (e+f x)\right ) \csc ^3(e+f x) \sec (e+f x) \left ((a+b) F_1\left (\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-b F_1\left (\frac{n+1}{2};n,1;\frac{n+3}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),\frac{(a-b) \tan ^2\left (\frac{1}{2} (e+f x)\right )}{a+b}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.668, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d\tan \left ( fx+e \right ) \right ) ^{n}}{a+b\sec \left ( fx+e \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \tan \left (f x + e\right )\right )^{n}}{b \sec \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (d \tan \left (f x + e\right )\right )^{n}}{b \sec \left (f x + e\right ) + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \tan{\left (e + f x \right )}\right )^{n}}{a + b \sec{\left (e + f x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \tan \left (f x + e\right )\right )^{n}}{b \sec \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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